Generalized Multipliers on Locally Compact Abelian Groups

Citation

Ford, Lawrence Charles (1974) Generalized Multipliers on Locally Compact Abelian Groups. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/NB31-1Y34. https://resolver.caltech.edu/CaltechETD:etd-10122005-082659

Abstract

NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. Let G be a locally compact Abelian group with dual [...], [...], and [...] supp [...] is compact}. Then for [...], the containments are proper if G is noncompact, and [...] is a dense, translation invariant subspace of [...] for [...]. Let [...] be a complex valued function defined on [...], and [...] = [...]. Suppose [...]. Define the operator, [...] by the equation [...] for each [...]. Then [...] is a module over M(G), [...] is a module homomorphism, and [...] is (p, q) closed. We call [...] a generalized (p, q) multiplier. The main results include: (1) Suppose T is an operator satisfying: (a) The domain D(T) is a translation invariant subspace of [...], and the range R(T) [...]; (b) D(T) [...]; (c) T is (p, q) closed, linear, and commutes with all translations; (d) C X T(C) is dense in [...]. Then T = [...] for some [...]. (2) The set of all generalized (p, q) multipliers, denoted [...], is a linear space, and the set of all generalized (p, p) multipliers, denoted [...], is an algebra containing [...] and contained in [...]. (3) If [...], then [...] is locally the transform of a bounded (p, q) multiplier. Further sections include a deeper study of [...], [...], and special results obtainable for compact G.

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